Find All The Zeros Of The Quadratic Function. Y = X 2 − 16 X + 64 Y = X^2 - 16x + 64 Y = X 2 − 16 X + 64 If There Is More Than One Zero, Separate Them With Commas. If There Are No Zeros, Select None.Zero(s): □ \square □ □ \square □ $\square, \square,

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this article, we will focus on finding the zeros of a quadratic function, which is a crucial step in solving quadratic equations.

What are Zeros of a Quadratic Function?

The zeros of a quadratic function are the values of the variable that make the function equal to zero. In other words, they are the solutions to the equation $ax^2 + bx + c = 0$. Finding the zeros of a quadratic function is essential in understanding the behavior of the function, as it helps us determine the points where the function intersects the x-axis.

The Quadratic Formula

The quadratic formula is a powerful tool for finding the zeros of a quadratic function. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The quadratic formula provides two solutions for the variable $x$, which are the zeros of the quadratic function.

Finding Zeros of a Quadratic Function: A Step-by-Step Guide

To find the zeros of a quadratic function, we can follow these steps:

1. Write the quadratic equation: Write the quadratic equation in the form $ax^2 + bx + c = 0$.
2. Identify the coefficients: Identify the coefficients $a$, $b$, and $c$ of the quadratic equation.
3. Plug in the values: Plug in the values of $a$, $b$, and $c$ into the quadratic formula.
4. Simplify the expression: Simplify the expression under the square root.
5. Solve for x: Solve for $x$ using the quadratic formula.

Example: Finding Zeros of a Quadratic Function

Let's consider the quadratic function $y = x^2 - 16x + 64$. To find the zeros of this function, we can follow the steps outlined above.

1. Write the quadratic equation: The quadratic equation is $x^2 - 16x + 64 = 0$.
2. Identify the coefficients: The coefficients are $a = 1$, $b = -16$, and $c = 64$.
3. Plug in the values: Plug in the values of $a$, $b$, and $c$ into the quadratic formula:

$x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(64)}}{2(1)}$

4. Simplify the expression: Simplify the expression under the square root:

$x = \frac{16 \pm \sqrt{256 - 256}}{2}$

5. Solve for x: Solve for $x$ using the quadratic formula:

$x = \frac{16 \pm \sqrt{0}}{2}$

$x = \frac{16}{2}$

$x = 8$

Therefore, the zero of the quadratic function $y = x^2 - 16x + 64$ is $x = 8$.

Conclusion

Finding the zeros of a quadratic function is a crucial step in solving quadratic equations. The quadratic formula is a powerful tool for finding the zeros of a quadratic function. By following the steps outlined above, we can find the zeros of a quadratic function and understand the behavior of the function.

Final Answer

The final answer is $8$.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In our previous article, we discussed how to find the zeros of a quadratic function using the quadratic formula. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations better.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for finding the zeros of a quadratic function. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula to find the zeros of a quadratic function?

A: To use the quadratic formula, follow these steps:

1. Write the quadratic equation: Write the quadratic equation in the form $ax^2 + bx + c = 0$.
2. Identify the coefficients: Identify the coefficients $a$, $b$, and $c$ of the quadratic equation.
3. Plug in the values: Plug in the values of $a$, $b$, and $c$ into the quadratic formula.
4. Simplify the expression: Simplify the expression under the square root.
5. Solve for x: Solve for $x$ using the quadratic formula.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is $ax + b = 0$, where $a$ and $b$ are constants, and $x$ is the variable.

Q: Can a quadratic equation have more than one solution?

A: Yes, a quadratic equation can have more than one solution. In fact, the quadratic formula provides two solutions for the variable $x$, which are the zeros of the quadratic function.

Q: How do I determine the number of solutions of a quadratic equation?

A: To determine the number of solutions of a quadratic equation, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the quadratic equation has two distinct solutions. If the discriminant is zero, the quadratic equation has one repeated solution. If the discriminant is negative, the quadratic equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$. If the discriminant is positive, the quadratic equation has two distinct solutions. If the discriminant is zero, the quadratic equation has one repeated solution. If the discriminant is negative, the quadratic equation has no real solutions.

Q: Can a quadratic equation have no real solutions?

A: Yes, a quadratic equation can have no real solutions. This occurs when the discriminant is negative, which means that the quadratic equation has complex solutions.

Q: How do I find the complex solutions of a quadratic equation?

A: To find the complex solutions of a quadratic equation, you can use the quadratic formula and take the square root of the negative discriminant. The complex solutions will be in the form $x = \frac{-b \pm \sqrt{-b^2 + 4ac}}{2a}$.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we provided a comprehensive Q&A guide to help you understand quadratic equations better. We discussed the quadratic formula, how to use it to find the zeros of a quadratic function, and how to determine the number of solutions of a quadratic equation.

Final Answer

The final answer is that quadratic equations are a powerful tool for solving problems in mathematics and other fields. By understanding the quadratic formula and how to use it, you can solve quadratic equations and find the zeros of a quadratic function.